1. Field of the Invention
This invention relates in general to white-light interferometry (WLI) techniques for surface characterization. In particular, it relates to a new method for extracting spectral information from an interference signal.
2. Description of the Related Art
The raw data from a white-light interferometer typically consist of an array of data sequence, the so-called correlograms. Each correlogram is a series of equally spaced light intensities recorded in a light detector as the objective lens of an optical interferometer scans across the zero optical path difference (OPD) point of the measurement light to produce various degrees of contrast. These light intensities result from two light beams interfering at a pixel of the detector. One beam is reflected from a sample surface and the other beam from a reference mirror. Those skilled in the art readily understand that many variations of this general description are possible; however, the optical principle is the same. FIG. 1 illustrates a typical correlogram.
In its physical nature, a correlogram is the product of wavefronts with different wavelengths in the source spectrum superposing at different phase shifts through the zero OPD position. In other words, light beams with different wavelength and intensity add together through the zero OPD position to form the correlogram, as illustrated in detail in FIGS. 2(a)-(c). FIG. 2(a) shows a Gaussian-shaped spectrum of a light source (i.e., the intensity distribution as a function of wavelength—λ1 to λ7 are illustrated). FIG. 2(b) shows the corresponding interference fringes produced by the different wavelengths in the spectrum (offset for clarity). FIG. 2(c) shows the superposition of all spectrum fringes, forming a correlogram with maximum contrast at the zero OPD position for the white light of FIG. 2(a) used to produce the correlogram.
For a simple reflective sample surface, the main parameters of interest in the correlogram are the envelope peak position z0 (i.e., the maximum fringe contrast position—see FIG. 1) and the corresponding fringe phase Φ. In as much as z0 is the zero OPD position for a given pixel, values of z0 from an array of pixels form a map z0(xi,yj) of the surface height. The corresponding phase values φ(xi,yj) form the so-called 2π (or phase) map.
For a more complex surface structure, such as a thin film stack, a structure with sub-wavelength patterns or roughness, composite materials, and so on, the reflected light spectrum and correspondingly the shape of the correlograms will change from the straightforward illustration of FIG. 1. In other words, the spectrum of the light captured at the detector, both in terms of spectral irradiance and spectral phase, will change to reflect the characteristics of the sample surface. Information regarding structural and material properties of the sample surface is embedded in the reflected light spectrum captured in the correlograms recorded at the detector. Thus, by analyzing the reflected spectral information, sample surface properties can be obtained. See, for example, Seung-Woo Kim and Gee-Hong Kim, who teach a method for profiling thin-film layers using spectral analysis (“Thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry,” Applied Optics, Vol. 38, No. 28, 1999, p. 5968) and T. Yatagai, who details a spectral analysis technique for profilometry as well as for measurement of material dispersion (“Recent Progresses in White Light Interferometry,” SPIE Vol. 2340, 1994, p. 338).
As detailed, for example, in U.S. Pat. No. 5,398,113, a typical way of obtaining spectral information (amplitude and phase) from a correlogram is through the use of Fourier transforms (FT). As generally described, the Fourier Transform S(k) of a correlogram is given by the relation
                              S          ⁡                      (            k            )                          =                              ∫                          -              ∞                                      +              ∞                                ⁢                                    C              ⁡                              (                z                )                                      ⁢                          exp              ⁡                              (                                  ⅈ                  ⁢                                                                          ⁢                  kz                                )                                      ⁢                                                  ⁢                          ⅆ              z                                                          (        1        )            where C(z) is the irradiance of the correlogram at position z and k is the spatial frequency of the interference fringe for a given spectral component of the light source. In WLI, C(z) values are collected at discrete scanning locations with j=1, 2, 3 . . . N. Therefore, the Fourier transform becomes a discrete Fourier transform (DFT),
                                          S            ⁡                          (              k              )                                =                                    ∑                              j                =                1                            N                        ⁢                                                  ⁢                                          C                ⁡                                  (                                      z                    j                                    )                                            ⁢                              exp                ⁡                                  (                                      ⅈ                    ⁢                                                                                  ⁢                                          kz                      j                                                        )                                                                    ;                                  ⁢                              (                                          z                j                            =                              jΔ                ⁢                                                                  ⁢                z                                      )                    .                                    (        2        )            The limit on spatial frequency for data acquisition purposes is given by the so called Nyquist critical frequency, i.e.
                                          k            C                    =                      1                          2              ⁢              Δ              ⁢                                                          ⁢              z                                      ,                            (        3        )            where Δz is the sampling interval (i.e., the scanning step size) for the data producing the correlogram. (Note that the sampling interval limits the maximum spatial frequency, or the bandwidth, not the frequency resolution).
As is well detailed in the prior art [see, for example, U.S. Pat. No. 7,106,454 (de Groot et al.) and U.S. Publication No. 2007/0091318 (Freishlad et al.)], an effective algorithm to calculate the DFT is the so-called fast Fourier transform (FFT), which transforms a sequence of N real numbers C1C2, . . . CN into a sequence of N complex numbers S1, S2, . . . SN(in WLI, the correlogram data sequence is usually real numbers), as follows:
                                          S            ⁡                          (                              k                m                            )                                =                                    ∑                              j                =                1                            N                        ⁢                                                  ⁢                                          C                ⁡                                  (                                      z                    j                                    )                                            ⁢                              exp                ⁡                                  (                                      ⅈ                    ⁢                                                                                  ⁢                                          k                      m                                        ⁢                                          z                      j                                                        )                                                                    ;                                  ⁢                              (                                          m                =                1                            ,              2              ,                              3                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                N                                      )                    .                                    (        4        )            
In the spatial frequency domain, the spectral interval (spectral resolution) km is limited by the total length of the number sequence C1, C2, . . . CN in the range of scan NΔz; i.e.,
                                          k            m                    =                      m                          N              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢              z                                      ;                                  ⁢                  m          =                                    -                              N                2                                      ⁢                                                  ⁢            …            ⁢                                                  ⁢                                          N                2                            .                                                          (        5        )            
That is, the spectral resolution Δk is given by
                              Δ          ⁢                                          ⁢          k                =                              1                          N              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢              z                                .                                    (        6        )            
FIGS. 3(a)-(c) illustrate this point by showing three correlograms and the corresponding FFT amplitudes for data-sequence lengths of 50, 100, and 200 frames, respectively (only positive frequencies are shown for simplicity of illustration, the negative frequencies being conjugate).
From the above and the illustration of FIG. 3 one can see that when FFT is used a desirable spatial frequency resolution can be obtained only with a large data window in the correlogram sequence even though the interference signals appear only in a small portion of the data sequence [see FIG. 3(c)]. This means that insufficient frequency information may result from a scan that does not cover a large enough window around the peak of modulation and, consequently, that a much longer scan than needed to generate the correlogram has to be carried out in order to obtain the desired spectral information. This feature, of course, increases the time required for performing the scan and for data acquisition and processing. Therefore, it would be very desirable to have an algorithm that did not require data acquisition beyond what is necessary for meaningful WLI analysis. The present invention achieves this objective using an in-phase/in-quadrature demodulation technique.